This paper addresses the problem of estimating entropy-regularized optimal transport (EOT) maps with squared-Euclidean cost between source and target measures that are subGaussian. In the case that the target measure is compactly supported or strongly log-concave, we show that for a recently proposed in-sample estimator, the expected squared $L^2$-error decays at least as fast as $O(n^{-1/3})$ where $n$ is the sample size. For the general subGaussian case we show that the expected $L^1$-error decays at least as fast as $O(n^{-1/6})$, and in both cases we have polynomial dependence on the regularization parameter. While these results are suboptimal compared to known results in the case of compactness of both the source and target measures (squared $L^2$-error converging at a rate $O(n^{-1})$) and for when the source is subGaussian while the target is compactly supported (squared $L^2$-error converging at a rate $O(n^{-1/2})$), their importance lie in eliminating the compact support requirements. The proof technique makes use of a bias-variance decomposition where the variance is controlled using standard concentration of measure results and the bias is handled by T1-transport inequalities along with sample complexity results in estimation of EOT cost under subGaussian assumptions. Our experimental results point to a looseness in controlling the variance terms and we conclude by posing several open problems.

翻译：本文研究具有平方欧氏代价的熵正则化最优传输（EOT）映射在源测度与目标测度均为次高斯情形下的估计问题。当目标测度具有紧支集或强对数凹性时，我们证明对于近期提出的样本内估计量，期望平方$L^2$-误差至少以$O(n^{-1/3})$的速率衰减（其中$n$为样本量）。对于一般次高斯情形，我们证明期望$L^1$-误差至少以$O(n^{-1/6})$的速率衰减，且两种情形下均对正则化参数呈现多项式依赖关系。尽管这些结果相较于源测度与目标测度均具有紧支集（平方$L^2$-误差以$O(n^{-1})$速率收敛）以及源测度为次高斯而目标测度具有紧支集（平方$L^2$-误差以$O(n^{-1/2})$速率收敛）等已知结果存在次优性，其重要性在于消除了对紧支集条件的要求。证明技术采用偏差-方差分解，其中方差项通过标准测度集中结果进行控制，偏差项则利用T1-传输不等式以及次高斯假设下EOT代价估计的样本复杂度结果进行处理。实验结果表明方差项控制存在松弛性，最后我们提出若干开放问题。